ON A RISK MODEL WITH A CONSTANT DIVIDEND AND DEBIT INTEREST

Transcription

1 Joral of Pr ad Appld ahas: Advas ad Applaos ol Nr Pags 87-4 ON A RSK ODEL WTH A CONSTANT DDEND AND DEBT NTEREST YUZHEN WEN Shool of ahaal Ss Qf Noral Uvrs Qf Shadog 765 P. R. Cha -al: wh@6.o Asra hs papr w osdr a rs odl wh d rs ad a osa dvdd arrr. W ass ha h la r pross s h grald Erlag () pross. For hs odl h o-grao fo h os of h dsod dvdd pas ad h Grr-Sh dsod pal fo ar vsgad. gral qaos ad gro-dffral qaos wh ra odar odos for h ar drvd.. rodo r ars h ss of asol r has rvd raral ao h aaral lrar. W ass ha wh h srpls s gav or h srr s o df h srr old orrow a ao of o qal o h df a a d rs for ( > ). awhl h srr wll rpa h ds oos fro hr pr o. Ths h srpls of h srr s drv dr h d rs for ahas Sj Classfao: 6J75. Kwords ad phrass: asol r Grr-Sh pd dsod pal fo o-grao fo grald Erlag pross. Ths wor was sppord Naoal Naral S Fodao of Cha (No. 779). Rvd Jaar ; Rvsd Jaar 6 Sf Advas Plshrs

2 88 YUZHEN WEN wh h srpls s gav. Th gav srpls a rr o a posv lvl. Howvr wh h gav srpls aas h lvl or s low h srpls s o logr al o posv as h ds of h srr a hs ar grar ha or qal o whh s h prs val a ha for all pr o avalal afr ha po. Asol r ors a hs o. Th asol r proal has sdd Grr [7] for h opod Posso odl wh h d ad h rd rs ras ar h sa. A losd for solo s gv h as of a poal la ao dsro. Dassos ad Erhs [] osdrd h r proal wh poal las sg h argal approah ad h hor of pws drs arov prosss. Erhs ad Shdl [5] osdrd h asol r proal for a or oplad rs odl. Th assd ha h opa a orrow o wh h srpls s gav ad rv rs for apal aov a ra lvl. Dso ad Egdo dos Rs [4] ad Zhag ad W [] dsssd h ff of rs o h gav srpls. Ca [] osdrd h Grr-Sh dsod pal fo a asol r. Wag ad Y [9] dsssd h o grao fo ad h os of h dsod dvdd pas. Gao ad Y [6] osdrd h o-grao fo a Sparr Adrs odl prrd dffso wh grald Erlag()-dsrd r-la s ad a hrshold dvdd srag. ovad Ca [] ad Gao ad Y [6] hs papr w sd h pd pal fo h o-grao fo ad h os of h dsod dvdd pas a asol r. L ( Ω F P) a opl proal spa oag all rado ojs dfd h followg. Sppos ha h srpls of a srr follows h rwal rs pross N ( ) U() Z S() (.) whr ad ar osas s h al srpls ad > s h ra of pr S N ( ) () Z s h aggrga las pross { Z :

3 ON A RSK ODEL WTH A CONSTANT 89 L } s a sq of dpd ad dall dsrd la ao ogav rado varals wh a oo dsro fo F. Th ordar rwal pross { N ( ) } dos h r of las p o wh N ( ) a{ : W W L W } whr h..d la wag s W hav a oo grald Erlag() dsro.. h W s ar dsrd as h s of dpd ad poall dsrd rado varals: W L L whr a hav dffr poal parars >. Frhror w ass ha { W } ad { } Z ar dpd ad E ( W ) > E. X W ow osdr a hrshold dvdd srag dvdds ar pad a a osa ra wh whvr U ( ) ( > ) ad o dvdds ar pad whvr U ( ). L { U ( ) } h odfd srpls of h srr a. Th a prssd as ( ) d ds( ) f U( ) ; du () d ds() f > U() ; (.) ( U ( )) d ds( ) f > U ( ). whr ( > ) dos h ao of apal h opa ras as a lqd rsrv ( ) s h dvdd ra ad ( > ) dos h for of rs for orrowd o. L T f { : U( ) } do h asol r of h srpls pross { U ( ) } whr T f asol r dos o or a f. W df h Grr-Sh pd dsod pal fo a asol r T E( ω( U U ) ( T ) U ) (.) T T

4 9 YUZHEN WEN whr ( w ) [ ) [ ) s a ogav fo whh dos h pal d a asol r U s h srpls dal T pror o asol r ad U T s h df a asol r s a posv osa whh a vwd as h rs for for h allao of h prs val of h pal ad ( A) s h daor fo of a v A. parlar h asol r proal s dod ( ) P( T U ). L D () h aggrga dvdds pad fro o. L T s D dd () s h prs val of all dvdds l asol r whr s h dso faor. W do h o-grag fo of D ( ) E[ ]. D hs papr w wll sd h pd dsod pal fo h o-grao fo ad h os of h dsod dvdd pas a asol r. So w g grodffral qaos ad gral qaos of h pd dsod pal fo a asol r. So w g h grodffral qaos of h o-grao fo ad h os of h dsod dvdd pas a asol r.. Th Grr-Sh Dsod Pal Fo S h Grr-Sh fo has dffr pahs for ad > w df ( ) > >. Slarl w l

5 ON A RSK ODEL WTH A CONSTANT 9 > >. Th followg hor provds gro-dffral qaos for h Grr-Sh fo. Thor.. Th Grr-Sh fo sasfs h followg gro-dffral qaos: Wh w hav [ ] A (.) wh > w hav [ ] A (.) ad wh [ ] A (.) wh h odar odos for L [ ] A (.4)

6 9 YUZHEN WEN [( ) ] [( ) ] (.5) [( ) ] [( ) ] (.6) whr A ω ( ) ad s h d opraor. Proof. For ooal ov w l ( ; L ) do h Grr-Sh fo wh h dsro of h r la s grald Erlag () pross ad ( ). B odog o h ad ao of h frs la ad dsog h pd vals o a h rs for wh w oa ( )[ ( ) ( ) w( ) ]d ()[ ( )( ) ( )( ) ( )( ) ( ) ( ) ( ( )( ) ) ( ( )( ) ) w( ( )( ) ( ( ) ( ))) ] d

7 ON A RSK ODEL WTH A CONSTANT 9 whr ad () s h oo ds fo of h la wag s W s. Wh > w hav ( )[ ( ) ( ) ( ) ( ) ( ) Wh > > w hav ( )[ ( ) ( ( ) ) ( ( ) ) ( ( ) ) w( ( ) ( ( ) ) ) ] d. h ( ) w h h h ( ) r ()[ ( h ( ) ) ( ) ]d ( ) ( ( ) ) ( ) ( ( ) ) ( ) ( ) Frs hagg varal w( ( ) ( )) ]. d wh rsp o fro o ( ) ad h hagg varal ( ) ( ) wh rsp o fro o w oa

8 YUZHEN WEN 94 d w. d w f w l h os. d w orovr frs hagg varal h h grals wh rsp o fro o ( s h solo of ) h ad h hagg varals h grals wh rsp o fro o w oa l d w

9 ON A RSK ODEL WTH A CONSTANT 95 d l. d w Th dffrag oh sds of ad wh rsp o ad s h rlao w oa h gro-dffral qaos: (.7). w (.8) Usg (.7) ad (.8) w hav [ ]. A Slarl w hav [ ] A [ ]. A

10 YUZHEN WEN 96 Th odar odos (.4)-(.6) a oad sg h sa args as ha [6 ]. Ths opls h proof. Corollar.. Th asol r proal sasfs h followg gro-dffral qaos: Wh w hav F (.9) wh > w hav F (.) ad wh F (.) wh h odar odos for L [ ] (.) [ ] [ ] (.)

11 ON A RSK ODEL WTH A CONSTANT 97 [ ] [ ]. (.4) Proof. f w l ad ω h F A ad h rsls follows. Codo (.) s ovos: f h A F whh oghr wh (.4) lds (.). f w ss for (.5) ad (.6) w g h (.) ad (.4). Corollar.. Cosdr h as wh ω ad F > whr > s a osa h sasfs h dffral qaos: Wh w hav [ ] (.5) wh > w hav [ ] (.6) ad wh [ ] [ ] j j j j (.7) wh h odar odos for L [ ] (.8)

12 YUZHEN WEN 98 [ ] [ ] (.9) [ ] [ ]. (.) Proof. Applg h opraor o (.) ad og ha d d A A w oa (.5). Eqaos (.6) ad (.7) ar prov aalogosl. Th proof of odar odos ar slarl o ha of Thor... o-grag Fo S has dffr pahs for ad > w df. ; ; W do h o of D [ ]. N D E Slar o also has dffr pahs for ad > w df

13 ON A RSK ODEL WTH A CONSTANT 99. ; ; parlar s h pao of D ha s [ ]. D E Thor.. Wh w hav F (.) wh > w hav F (.) wh w hav F (.) wh h odar odos for L

14 YUZHEN WEN ( ) (.4) ( ) ( ) ( ). (.5) (.6) Proof. () Wh w do ( ) ( S ) ad ( ) ( ) for L whr S S L. S { : L } ar poall dsrd r.v s w osdr h fsal rval [ S S d]. B odog o h ad ao of h frs la ad whhr h la ass asol r w oa d ( ) ( d) ( d ) d d ( d ) o( d). (.7) Talor s paso gvs d ( d ) ( ) ( ) ( ) d o( d). (.8) Ssg (.8) o (.7) dvdg oh sds of (.7) lg ad rarragg w oa ( ) ( ) ( ) ( ).

15 ON A RSK ODEL WTH A CONSTANT Tha s ( ) ( ) ( ) ( ) (.9) whr L. Now l w hav d ( ) ( d) ( d ) d ( ) ( ) ( ) F ( ) o( d). Tha s ( ) ( v ) ( v) ( ) F ( ). (.) Ssg (.) o (.9) w hav (.). () Usg h args slar o hos sd (.6) ad (.7) w hav (.) ad (.). Aordg o [6] w hav ( ) ( ) j L ad j j ( ) j ( ) j j L whh oghr wh (.9) lds (.4). Slarl w a g h odar odos of (.5) ad (.6).

16 YUZHEN WEN Thor.. Wh [ ] (.) wh > [ ] (.) ad wh [ ] (.) wh h odar odos for L. Proof. B h dfos of ad w oa.! (.4)

17 ON A RSK ODEL WTH A CONSTANT Ssg (.) o (.) (.) ad (.) rspvl ad oparg h offs of N ldg h rsls. Corollar.. Cosdr h as wh > F h sasfs h dffral qaos: Wh [ ] (.5) wh > w hav [ ] (.6) ad wh [ ] [ ] j j j j (.7) wh h odar odos for L. Proof. Th proof s slar o ha of Corollar..

18 4 YUZHEN WEN Aowldgs Th ahor wold l o ha h aoos rfrs for hr osrv ad sghfl sggsos ad os o h prvos vrso of hs papr. Rfrs [] S. Asss R Proals World Sf Sgapor. [] J. Ca O h val of asol r wh d rs Advas Appld Proal 9 (7) [] A. Dassos ad P. Erhs argals ad sra rs Sohas odls 5 (989) 8-7. [4] E. Dso ad Egdo dos Rs Th ff of rs o gav srpls sra: ahas ad Eoos (997) -6. [5] P. Erhs ad H. Shdl R sao for a gral sra odl Advas Appld Proal 6 (994) [6] H. L. Gao ad C. C. Y Th prrd Sparr Adrs odl wh a hrshold dvdd srag J. Cop. Appl. ah. (8) [7] H. Grr Dr Eflss vo Zs af d Rwahrshlh Bll. Wss. Asso. Aar 7 (97) 6-7. [8] T. Rols H. Shdl. Shd ad J. Tgls Sohas Prosss for sra ad Fa Wl Chhsr U.K [9] C. W. Wag ad C. C. Y Dvdd pas h lassal rs odl dr asol r wh d rs Appld Sohas odl Bsss ad dsr 5 (9) [] H. Yag Z.. Zhag ad C.. La O h val of asol r for a llar opod Posso odl dr rs for Sa. Pro. L. 78 (8) [] C. S. Zhag ad R. W O h dsro h srpls of h D-E odl pror o ad r sra: ahas ad Eoos 4 () 9-. g